394 CHAPTER 6 Trigonometric Functions 96. Designing a Water Sprinkler An engineer is asked to design a water sprinkler that will cover a field of 100 square yards that is in the shape of a sector of a circle of radius 15 yards. Through what angle should the sprinkler rotate? 97. Windshield Wiper The arm and blade of a windshield wiper have a total length of 34 inches. If the blade is 25 inches long and the wiper sweeps out an angle of 120 ,° how much window area can the blade clean? 98. Windshield Wiper The arm and blade of a windshield wiper have a total length of 30 inches. If the blade is 24 inches long and the wiper sweeps out an angle of 125 ,° how much window area can the blade clean? 99. Motion on a Circle An object is traveling on a circle with a radius of 5 centimeters. If in 20 seconds a central angle of 1 3 radian is swept out, what is the angular speed of the object? What is its linear speed? 100. Ferris Wheels A neighborhood carnival has a Ferris wheel with a radius of 30 feet.You measure the time it takes for one revolution to be 70 seconds.What is the linear speed (in feet per second) of this Ferris wheel? What is the angular speed in radians per second? 101. Amusement Park Ride A gondola on an amusement park ride, similar to the Spincycle at Silverwood Theme Park, spins at a rate of 13 revolutions per minute. If the gondola is 25 feet from the ride’s center, what is the linear speed of the gondola in miles per hour? 102. Amusement Park Ride A centrifugal force ride, similar to the Gravitron, spins at a rate of 22 revolutions per minute. If the diameter of the ride is 13 meters, what is the linear speed of the passengers in kilometers per hour? 103. Wind Turbine A wind turbine located in Rotterdam, Netherlands, has a rotor diameter of 222 meters. If the blades turn at a rate of 14 revolutions per minute, what is the linear speed of the blade tip, in km hr? 104. Turntable Vinyl turntables are making a comeback among audiophiles. Suppose a turntable is playing a vinyl record at 45 rotations per minute. If a record has a 7 inch diameter, what is the linear speed, in miles per hour, of a point 3 inches from the center? Round your answer to two decimal places. 105. Bicycle Wheels The diameter of each wheel of a bicycle is 26 inches. If you are traveling at a speed of 35 miles per hour on this bicycle, through how many revolutions per minute are the wheels turning? 106. Car Wheels The radius of each wheel of a car is 15 inches. If the wheels are turning at the rate of 3 revolutions per second, how fast is the car moving? Express your answer in inches per second and in miles per hour. 107. Photography If the viewing angle for a 600 mm lens is 4 6 , ° ′ use arc length to approximate the field width of the lens at a distance of 860 feet. 108. Photography If the viewing angle for an 800mm lens is 1 42 , ° ′ use arc length to approximate the field width of the lens at a distance of 920 feet. In Problems 109–110, the latitude of a location L is the angle formed by a ray drawn from the center of Earth to the equator and a ray drawn from the center of Earth to L. See the figure. South Pole L Equator North Pole u 109. Linear Speed on Earth Earth rotates on an axis through its poles. The distance from the axis to a location on Earth at 30° north latitude is about 3429.5 miles. Therefore, a location on Earth at 30° north latitude is spinning on a circle of radius 3429.5 miles. Compute the linear speed on the surface of Earth at 30° north latitude. 110. Linear Speed on Earth Earth rotates on an axis through its poles. The distance from the axis to a location on Earth at 40° north latitude is about 3033.5 miles. Therefore, a location on Earth at 40° north latitude is spinning on a circle of radius 3033.5 miles. Compute the linear speed on the surface of Earth at 40° north latitude. 111. Speed of the Moon The mean distance of the moon from Earth is 2.39 10 miles. 5 × Assuming that the orbit of the moon around Earth is circular and that 1 revolution takes 27.3 days, find the linear speed of the moon. Express your answer in miles per hour. 112. Speed of Earth The mean distance of Earth from the Sun is 9.29 10 miles. 7 × Assuming that the orbit of Earth around the Sun is circular and that 1 revolution takes 365 days, find the linear speed of Earth. Express your answer in miles per hour. 113. Pulleys Two pulleys, one with radius 2 inches and the other with radius 8 inches, are connected by a belt. (See the figure.) If the 2-inch pulley is caused to rotate at 3 revolutions per minute, determine the revolutions per minute of the 8-inch pulley. [Hint: The linear speeds of the pulleys are the same; both equal the speed of the belt.] 8 in. 2 in. 114. Pulleys Two pulleys, one with radius r1 and the other with radius r ,2 are connected by a belt. The pulley with radius r1 rotates at 1ω revolutions per minute, whereas the pulley with radius r2 rotates at 2ω revolutions per minute. Show that r r 1 2 2 1 ω ω = Use this result to rework Problem 113. Credit: Taigi/Shutterstock
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